Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-16T15:24:55.317Z Has data issue: false hasContentIssue false
  • English
  • Français

Sets of finite sets satisfying union conditions

Published online by Cambridge University Press:  26 February 2010

David E. Daykin  and
Peter Frankl
David E. Daykin
Affiliation:
University of Reading, Reading. RG6 2AX
Peter Frankl
Affiliation:
C.N.R.S., Paris, France.
Get access

Abstarct

Let ℱ denote a set of subsets of X = {1, 2,…, n). Let deg(i) be the number of members of ℱ containing i and val(ℱ) = min {deg (i): i ∈ X). Suppose no k members of ℱ have union X. We conjecture val(ℱ) ≤ 2n-k-1 for k ≥ 3. This is known for n ≤ 2k and we prove it for k ≥ 25. For k = 2 an example has val(ℱ) > 2n-2(l−n-0·651) and we prove val(ℱ) ≤ 2n-2(1–n-1). We also prove that if the union of k sets one from each of ℱ1,…, ℱk has cardinality at most n – t then min {cardinality ℱj} < 2nαt where αk = 2α − 1 and ½ < α < 1.

Type
Research Article
Information
Mathematika , Volume 29 , Issue 1 , June 1982 , pp. 128 - 134
Copyright
Copyright © University College London 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Brace, A. and Daykin, D. E.. Cover theorems for finite sets, I, II, III. Bull. Austral. Math. Soc., 5 (1971), 197–202 and 6 (1972), 19–24 and 6 (1972), 417–433.CrossRefGoogle Scholar
2.Brace, A. and Daykin, D. E.. A finite set covering theorem IV. Coll. Math. Soc. Jänos Bolyai, Keszthey, (1973), 199–203.Google Scholar
3.Brace, A. and Daykin, D. E.. Sperner type theorems for finite sets. In Combinatorics Proc. Conf. Oxford 1972 (Inst. of Mathematics and its Applications, 1972), 18–37.Google Scholar
4.Daykin, D. E.. Minimum subcover of a finite set. Amer. Math. Monthly, 85 (1978), 766.CrossRefGoogle Scholar
5.Erdős, P., Ko, C. and Rado, R.. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford. II, 12 (1961), 313–320.CrossRefGoogle Scholar
6.Frankl, P.. Families of finite sets satisfying an intersection condition. Bull. Austral. Math. Soc., 15 (1976), 73–79.CrossRefGoogle Scholar
7.Frankl, P.. Families of finite sets satisfying a union condition. Discrete Math., v (1979), 111–118.CrossRefGoogle Scholar
8.Katona, G.. Intersection theorems for finite sets. Acta. Math. Acad. Sci. Hungar., 15 (1964), 329–337.CrossRefGoogle Scholar
9.Feller, W.. An introduction to probability theory and its applications, Vol II (John Wiley, New York, 1966; 2nd edition 1971).Google Scholar